Common mistakes when evaluating limits of piecewise functions

📚 Understanding Piecewise Functions and Limits

A piecewise function is a function defined by multiple sub-functions, each applying to a certain interval of the main function's domain. Evaluating limits of piecewise functions requires careful consideration of which sub-function applies as you approach the point of interest. This guide will walk you through the common mistakes and how to avoid them.

📜 Definition

A piecewise function is defined as:

$f(x) = \begin{cases} f_1(x) & \text{if } x < a \\ f_2(x) & \text{if } x \geq a \end{cases}$

📅 Historical Context

Piecewise functions have been used implicitly for centuries, but a formal understanding and notation developed alongside the growth of calculus in the 17th and 18th centuries. They became essential for modeling situations with abrupt changes or different behaviors across domains, finding applications in physics and engineering.

🔑 Key Principles

• 🔍 Identify the relevant sub-function: Determine which part of the piecewise function applies to the limit you are evaluating. If you are finding the limit as $x$ approaches $a$, you need to consider the functions defined around $x = a$.
• ⬅️ Left-hand limit: The limit as $x$ approaches $a$ from the left (denoted as $x \to a^-$). Use the sub-function defined for $x < a$.
• ➡️ Right-hand limit: The limit as $x$ approaches $a$ from the right (denoted as $x \to a^+$). Use the sub-function defined for $x > a$.
• 🤝 Existence of the limit: For the limit to exist at $x = a$, the left-hand limit and the right-hand limit must be equal, i.e., $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$.
• 🚧 Continuity: A piecewise function is continuous at $x = a$ if the limit exists at $x = a$ and is equal to the function's value at $x = a$, i.e., $\lim_{x \to a} f(x) = f(a)$.

🚫 Common Mistakes

• 🍎 Ignoring the correct interval: Using the wrong sub-function when evaluating the limit. Always check which interval contains the value $x$ is approaching.
• 🧮 Not checking one-sided limits: Assuming the limit exists without evaluating both the left-hand and right-hand limits.
• 📈 Incorrectly evaluating limits: Making mistakes in evaluating the limits of the individual sub-functions (e.g., algebraic errors, misapplication of limit laws).
• ♾️ Ignoring undefined points: Not recognizing points where the function or its sub-functions are undefined.
• 📊 Misinterpreting the graph: Failing to understand the graphical representation of the piecewise function, leading to errors in determining limits.

💡 Tips to Avoid Mistakes

• 📝 Write it out: Explicitly write down the left-hand and right-hand limits separately.
• ✔️ Double-check: Verify that you're using the correct sub-function for each one-sided limit.
• 📐 Graph it: Sketch a graph of the piecewise function to visualize its behavior near the point of interest.
• 🧮 Simplify: Simplify each sub-function as much as possible before evaluating the limit.

🌍 Real-world Examples

Example 1:

Consider the function:

$f(x) = \begin{cases} x^2 & \text{if } x < 1 \\ 2x & \text{if } x \geq 1 \end{cases}$

To find $\lim_{x \to 1} f(x)$, we check the one-sided limits:

$\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} x^2 = 1$

$\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} 2x = 2$

Since the left-hand limit (1) is not equal to the right-hand limit (2), the limit $\lim_{x \to 1} f(x)$ does not exist.

Example 2:

Consider the function:

$g(x) = \begin{cases} x + 1 & \text{if } x < 2 \\ 3 & \text{if } x \geq 2 \end{cases}$

To find $\lim_{x \to 2} g(x)$, we check the one-sided limits:

$\lim_{x \to 2^-} g(x) = \lim_{x \to 2^-} (x + 1) = 3$

$\lim_{x \to 2^+} g(x) = \lim_{x \to 2^+} 3 = 3$

Since the left-hand limit (3) is equal to the right-hand limit (3), the limit $\lim_{x \to 2} g(x) = 3$. Also, $g(2) = 3$, so the function is continuous at $x = 2$.

📝 Practice Quiz

Evaluate the following limits for the given piecewise functions:

1. $f(x) = \begin{cases} 3x - 1 & \text{if } x < 0 \\ x^2 + 2 & \text{if } x \geq 0 \end{cases}$. Find $\lim_{x \to 0} f(x)$.
2. $g(x) = \begin{cases} 4 - x & \text{if } x < 3 \\ 2x - 5 & \text{if } x \geq 3 \end{cases}$. Find $\lim_{x \to 3} g(x)$.
3. $h(x) = \begin{cases} x^3 & \text{if } x < -1 \\ 2x + 3 & \text{if } x \geq -1 \end{cases}$. Find $\lim_{x \to -1} h(x)$.

✅ Conclusion

Evaluating limits of piecewise functions requires careful attention to detail, especially when determining which sub-function applies and when checking one-sided limits. By understanding the key principles and avoiding common mistakes, you can confidently tackle these problems. Remember to always check the definition of the piecewise function and the point at which you are evaluating the limit.